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2145-391 Aerospace Engineering Laboratory I. Measurement Measurement Errors / Models Measurement Problem and The Corresponding Measurement Model Measure with Single Instrument:Single Sample / Multiple Samples Measure with Multiple Instruments:Single Sample / Multiple Samples

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2145-391 Aerospace Engineering Laboratory I

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2145 391 aerospace engineering laboratory i

2145-391 Aerospace Engineering Laboratory I

  • Measurement

  • Measurement Errors / Models

    • Measurement Problem and The Corresponding Measurement Model

      • Measure with Single Instrument:Single Sample / Multiple Samples

      • Measure with Multiple Instruments:Single Sample / Multiple Samples

  • Uncertainty of A Measured Quantity (VS Uncertainty of A Derived Quantity – Next Week)

  • Measurement Statement

    • Single Sample Measurement

    • Multiple Samples Measurement


Some details of contents

Some Details of Contents

  • Where are we?

  • Measured Quantity VSDerived Quantity

  • Objectives and Motivation

  • Deterministic Phenomena VS Random Phenomena

  • Measurement Problems, Measurement Errors, Measurement Models

  • Population and Probability

    • Probability Distribution Function (PDF)

    • Probability Density Function (pdf)

    • Expected Value

    • Moments

  • Sample and Statistics

    • Sample Mean and Sample Variance

  • Interval Estimation

  • Terminologies for Measurement: Bias, Precise, Accurate

  • Error VS Uncertainty

  • Measurement Statement

    • Measured Variable as A Random Variable

    • Uncertainty of A Measured Quantity [VS Uncertainty of A Derived Quantity – Next Week]

    • Measurement Statement

  • Experimental Program: Test VS Sample

  • Some Uses of Uncertainty


Where are we on drd

  • Week 1: Knowledge and Logic

  • Week 2:

  • Structure and Definition of an Experiment

  • DRD/DRE

bottommost level

bottommost level

Week 3: Instruments

Week 4: Measurement and Measurement Statement:

Where are we on DRD?


Measured quantity vs derived quantity

Measured Quantity VSDerived Quantity

Recall the difference between

Measured Quantityy

(numerical value of yis determined

from measurement with an instrument)

VS

Derived Quantityy

(numerical value of y is determined

from a functional relation)

In this period we focus first on measured quantity.


Objectives

Objectives

Measurement Statement:

If you

  • know what this measurement statement says (regarding measurement result),

  • know its use [what good is if for, and when to use it]

  • know and understand its underlying ideas [why should we report a measurement result with this statement],

  • (know, understand, and) know how to report a measurement result of a measured quantity with this measurement statement for the cases of

    • A single-sample measurement

    • A multiple-sample measurement

      we can all go home.

Activity: Class Discussion on the above


Class activity and discussion

Right. I can just simply close my eyes and bet against

Class Activity and Discussion

  • 10 students come up and measure the resistance of a given resistor.

  • Then, report the measurement result.

    Discussions

  • Do they get the same result?

  • If not, what is, and who gets, the ‘correct’ value then?

Wanna bet500 bahts? [on whom, or which value]

  • If the next 10 of your friends come up and measure the resistance, and

    less than 9 out of 10 of themdo not get the value you bet on,

    you lose and give me 500 bahts. Otherwise, I win.

    Or

  • I’ll let you set the term of our bet.[Of course, I have to agree on the term first.]

    What term should our bet be then? [Make it reasonable and “bettable.”]


Motivation

Motivation

Given that there are some (random) variations in repeated measurements,

how should we report a measurement result so that it makes some usable sense?

The Why

The Use

The How

of Measurement Statement


2145 391 aerospace engineering laboratory i

Deterministic Phenomena VS Random Phenomena

Deterministic Variable VS Random Variable


Deterministic phenomena determinsitic variable

In reality, however, chances are that we will not have that exact position

Deterministic Phenomena Determinsitic Variable

Deterministic Phenomena

(1) The state of the system at time t, and

(2) its governing relation,

deterministically determine the state of the system – or the value of the deterministic variable y -

at any later time.

Example: Free fall (and Newton’s Second law)


Random phenomena random or statistical experiment random variable

Random Phenomena [Random or Statistical Experiment]Random Variable

A random or statistical experiment is an experiment in which [1]

  • all outcomes of the experiment are known in advance,

  • any realization (or trial) of the experiment results in an outcome that is not known in advance, and

  • the experiment can be repeated under nominally identical condition.

    [1]Rohatgi, V. K., 1976, An Introduction to Probability Theory and Mathematical Statistics, Wiley, New York, p. 20.


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Class Activity:Discussion

Is tossing a coin a random experiment?

  • All possible outcomes are H and T and nothing more.

  • For any toss, we cannot know/predict the outcome in advance.

    HorT?

    3.Care can be taken to repeat the toss under nominally identical condition.


2145 391 aerospace engineering laboratory i

Measurement:

  • All possible outcomes are known in advance, e.g.,

  • For any one realization w, we cannot know/predict the exact outcome with certainty in advance.

  • Care can be taken to repeat the measurement under nominally identical condition.

Class Activity:Discussion

Is measurement a random experiment?


2145 391 aerospace engineering laboratory i

Measurement Problems

Measurement Errors

Measurement Models


Measurement problems measurement errors and the corresponding measurement models

Measurement Problems,Measurement Errors,and The Corresponding Measurement Models

  • Measure with a single instrument

    • Single Sample

    • Multiple Samples

  • Measure with multiple instruments

    • Single Sample

    • Multiple Samples

  • Other models are possible, depending upon the nature of errors considered.


  • Measurement problem measure with a single instrument error at measurement i

    X

    Measured value of X at reading/sample i :

    True value ofX :

    Total measurement error for reading i :

    Measurement Problem: Measure with A Single Instrument: Error at Measurement i

    Error at measurement i

    Measurement i

    We never know the true value.


    Decomposition of error systematic bias error vs random precision error

    X

    Systematic/Bias error:

    Constant. Does not change with realization.

    Random/Precision error for reading i,

    Randomly varying from one realization to another.

    Measurement model

    Decomposition of ErrorSystematic/Bias Error VS Random/Precision Error

    Statistical Experiment:

    Repeated measurements under nominally identical condition

    Measurement i


    Measurement problem measure with a single instrument measurement model 1

    Measurement Model 1:

    The ith measured value

    The ith observed value of a random variableX

    True value

    [Can never be known for certainty, not a random variable]

    Systematic / Bias error

    [Constant. Does not change with realization,not a random variable]

    Random / Precision error

    [Randomly varying from one realization to another, a random variable]

    Measurement Problem: Measure with A Single Instrument: Measurement Model 1


    2145 391 aerospace engineering laboratory i

    Frequency of occurrences

    the population distribution

    of measured valueX

    X

    Measured Value [Random Variable] How to Describe/Quantify A Random Variable: Distribution of Measured Value [Random Variable]

    Statistical Experiment:

    Repeated measurements under nominally identical condition

    Measurement i

    Repeated measurements under nominally identical condition

    Description of Deterministic Variable:

    State the numerical value of the variable under that condition.

    Description of Random Variable

    Since it randomly varies from one realization to the next

    [even under the same nominal condition and we cannot predict its value exactly in advance],

    a meaningful way to describe it is by describing its probability (distribution).

    VS


    Probability and statistics

    Population

    Sample

    Probability and Statistics

    Probability – Deductive reasoning

    Given properties of population,

    extract information regarding a sample.

    Statistics – Inductive reasoning

    Given properties of a sample,

    extract information regarding the population.


    2145 391 aerospace engineering laboratory i

    Population

    Sample

    Population and Probability

    Probability – Deductive Reasoning

    Given properties of population,

    extract information regarding a sample.


    Probability distribution function pdf

    Some Properties

    x

    The probability of an event is the value of

    Probability Distribution Function (PDF)


    Probability density function pdf

    Some Properties

    Area

    or

    = Area under the pdf curve from to x.

    x

    Probability Density Function (pdf)

    Thin / high VS Wide / low


    Probability of an event

    x

    Probability of An Event

    PDF FX(x)

    pdf fX(x)

    x

    • The probability of an event is

      • 1. the area under the fX(x) curve from to x,

      • 2. the value of FX(x).


    Probability of an event1

    Area

    x2

    x1

    Probability of An Event

    FX(x2)

    PDF FX(x)

    pdf fX(x)

    FX(x2) - FX(x1)

    FX(x1)

    x

    • The probability of an event is

      • 1. the area under the fX(x) curve from x1 to x2,

      • 2. the value of [FX(x2) - FX(x1)].


    Example of some pdf uniform density function

    PDF: U(x;-1,1)

    PDF: U(x;-2,2)

    U(x;-1,1)

    U(x;-2,2)

    x

    Example of Some pdf: Uniform Density Function

    Function U(x) with parameters a and b:

    Thin / high VS Wide / low


    Example of some pdf normal density function

    x

    Example of Some pdf: Normal Density Function

    Function N(x) with parameters m and s 2:

    PDF: N(x;0,1)

    PDF: N(x;0,2)

    N(x;0,1)

    N(x;0,2)


    Example of some pdf student s t density function

    x

    Example of Some pdf: Student’s t Density Function

    Function t(x) with parameter n :

    PDF: t(x; 20)

    PDF: t(x; 5)

    t(x; 5)

    t(x; 20)


    Example of some pdf chi squared density function

    x

    Example of Some pdf: Chi-Squared Density Function

    Function c2(x) with parameter n :

    c 2(x; 5)

    c 2(x; 10)

    c 2(x; 15)

    c 2(x; 20)


    Expected values of a random variable

    Expected Values of A Random Variable

    Definition: Expected Value of A Random Variable X

    The expected value (or the mathematical expectation or the statistical average) of a continuous random variable X with a pdf fX(x) is defined as

    Definition:Expected Value of A Function of A Random Variable X

    Let Y = g(X) be a function of a random variable X, then Yis also a random variable,

    and we have

    However, we can also calculate E(Y) from the knowledge of fX(x) without having to refer to fY(y)as


    Moments of a pdf

    Moments of A pdf

    Definition:Moment About The Origin

    The rth-ordermoment about the origin (of a df) of X, if it exists, is defined as

    where r = 0,1,2,….

    Note that this is the rth-order moment of area under fX(x)about the origin.

    Definition:Central Moment

    The rth-order central momentof a df of X, if it exists, is defined as

    where r = 0,1,2,….and E[X] = mX.

    Note that this is the rth-order moment of area under fX(x)about mX.


    Interpretations of moments

    fX (x)

    x

    dx

    dA

    dA

    Interpretations of Moments

    moment arm for Mr = x(r)

    moment arm for mr = (x-mX)(r)

    Area dA = fX(x)dx

    NOTE: Due to the rth-power of the arm length, the values of fX (x) at further distance from the center (origin or mX) relatively contribute more to the moment than those at closer to the center.

    x

    x - mx

    mX


    Some properties of m r and m r

    Some Properties of Mr and mr

    Properties of Origin Moment Mr

    Moment order 0:

    Moment order 1 (Mean of rv X):

    Moment order 2

    Properties of Central Moment mr

    Moment order 0:

    Moment order 1:

    Moment order 2:

    (Variance sX2)

    mXis the location of the centroid of the pdf.

    sX2is a measure of the width of the pdf.


    2145 391 aerospace engineering laboratory i

    Population

    Sample

    Population Mean

    Population Variance

    Sample Mean

    Sample Variance

    How close is to in some sense?

    Sample and Statistics

    Since we do not know the properties of the population ,

    we want to estimate them with the statistics drawn from a sample.

    Statistics – Inductive reasoning


    Sample mean and sample variance

    Sample Mean and Sample Variance

    Definition

    Let X1, X2, …, Xnbe a random sample from a distribution function fX(x).

    Then, the following statistics are defined.

    Sample Mean:

    Sample Variance:

    Sample Standard Deviation:

    Sample mean, sample variance, and sample standard deviation are statistics, hence, random variables, not simply numbers.

    Unbiased estimator of mX.

    Unbiased estimator of .


    2145 391 aerospace engineering laboratory i

    Interval Estimation

    Assume X is a random variable whose pdf is normal and

    Let (X1, X2, …, Xn)be an iid random sample from

    • Interval Estimation: Probability Distributions of Random Variables


    Convention on a

    pdf

    pdf

    Normal and Students t

    Chi Squared

    Area = a/2

    Area = a/2

    Convention on a


    Interval estimation

    Interval Estimation

    Theorem 1: Standard Normal Random Variable

    If , then .

    Zis called a standard normal random variable.

    In addition, we have

    or

    whereza/2 denotes the value on the z axis for which a/2of the area under the z curve lies to the right of za/2.

    magnitude of the deviation/distance

    from X to mX, or from mX to X.


    2145 391 aerospace engineering laboratory i

    The probability that deviates from no more than ( times ) is

    .


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    Theorem 2:Distribution for A Random Variable

    Let (X1, X2, …, Xn)be a sample from .

    Then, the random variable has

    Hence,

    or


    2145 391 aerospace engineering laboratory i

    The probability that deviates from no more than ( times ) is .


    2145 391 aerospace engineering laboratory i

    Theorem 3:Distribution for A Random Variable

    (Student’s t Distribution)

    Let (X1, X2, …, Xn)be a sample from .

    Then, the random variable has

    that is, T has a Student’s t distribution with degree of freedom n = n -1.

    Hence,

    or


    2145 391 aerospace engineering laboratory i

    The probability that deviates from no more than ( times ) is .


    2145 391 aerospace engineering laboratory i

    One More Sample from Previously Drawn n Samples (Large Sample Size Approximate, n large)

    Let (X1, X2, …, Xn)be a sample from and

    be the sample variance of this sample.

    Let be an additional single sample drawn from .

    Then, the random variable has

    that is, T has a Student-t distribution withdegree of freedom n = n-1.

    Hence,

    or


    2145 391 aerospace engineering laboratory i

    The probability that deviates from no more than ( times ) is .


    Summary of interval estimation scheme diagram

    Summary of Interval Estimation Scheme Diagram


    2145 391 aerospace engineering laboratory i

    Students t

    pdf

    Area = a/ 2

    Interval Estimation

    Assume X is a random variable whose pdf is normal and

    Let (X1, X2, …, Xn)be an iid random sample from

    • Interval Estimation: Probability Distributions of Random Variables


    2145 391 aerospace engineering laboratory i

    Terminologies for Measurement

    Bias

    Precise

    Accurate


    Terminologies for measurement bias precise and accurate

    X

    X

    XTrue , m X

    XTrue, m X

    XTrue,

    m X

    X

    X

    XTrue

    m X

    Terminologies for Measurement: Bias, Precise, and Accurate

    Frequency of occurrences

    Biased + Imprecise  Inaccurate

    Biased + Precise  Inaccurate

    Unbiased + Imprecise  Inaccurate

    Unbiased + Precise  Accurate


    2145 391 aerospace engineering laboratory i

    Error VS Uncertainty


    Terminologies error vs uncertainty

    Terminologies: Error VS Uncertainty

    • Error

      If the error is known for certainty, (it is the duty of the experimenter to) correct it and it is no longer an error.

    • Uncertainty

      For error that is not known for certainty, no correction scheme is possible to correct out these errors.

      In this respect, the termuncertainty is more suitable.

    • The two terms sometimes – if not often – are used without strictly adhere to this. Nonetheless, the above should be recognized.


    2145 391 aerospace engineering laboratory i

    Measurement Statement

    Measured Variable as A Random Variable

    Uncertainty of A Measured Quantity

    [VS Uncertainty of A Derived Quantity – Next Week]

    Measurement Statement


    Measurement model 1 bias vs precision random scatter

    Sample width

    Population width

    ei

    Area = 1- a

    Bias

    i

    Measurement Model 1: Bias VS Precision/Random (Scatter)

    For repeated measurements under nominally identical condition:

    • Bias:results in the deviation of the (population) mean from the true value.

    • Precision/Random:results in the scatter in datain a set of repeated measurements.

      It is viewed and quantified as

      the band width of the scatter

      not absolute position.


    Measurement model 2 replacement concept bias error as random variable

    One realization of random error

    +BX: Bias Uncertainty

    e i

    bj: One realization of bias error.

    XTrue

    +PX:Random Uncertainty

    i

    Measurement Model 2 :Replacement Concept: Bias Error as Random Variable

    If the measurement instrument identity is changed, the bias is changed and is considered a random variable.

    Xj,i=ithrealization of instrumentj


    Uncertainty of a measured quantity

    Uncertainty of A Measured Quantity


    U x root sum square rss of b x and p x

    1

    Measurement statementi

    UX = Root-Sum-Square (RSS) of BX and PX

    Bias and Precision/Random uncertainties are combined with

    root-sum-square (rss) method.


    Estimating b x and p x

    • Bias Uncertainty

      Bias Uncertainty = Root-sum-square (RSS) of elemental error sources.

      To the very least, it is the uncertainty of the instrument itself, e.g.,

    • Precision Uncertainty

      Report:

      Single value:

      Average value:

    Estimating BX and PX


    2145 391 aerospace engineering laboratory i

    Measurement Statement as An Interval Estimate


    2145 391 aerospace engineering laboratory i

    Experimental Program:

    Test VS Sample


    Terminologies test vs measurement reading sample experiment

    Terminologies: Test VS Measurement/Reading/SampleExperiment

    Test:The word is associated with the evaluation of DRD/DRE, or r.

    One Test = One Evaluation of DRE  One r

    Measurement, Reading, Sample: The word is associated with Xireading from the individual measurement system i.

    One Sample = One Measurement of Xi  One Xi

    ST/SS

    Single Test/Single Sample

    Single

    Single X Single r

    Single r

    Single r

    ST/MS

    Single Test/Multiple Sample

    Single r

    Multiple

    Average X Single r

    Single r

    MT/SS

    Multiple Test/Single Sample

    Single

    Average r

    Single X Single r

    Multiple r

    MT/MS

    Multiple Test/Multiple Sample

    Multiple

    Average X Single r

    Multiple r

    Average r


    Data analysis for various types of experiment

    Average overrk

    MT/SS

    Average over(Xi)k

    Average overrl

    MT/MS

    Data Analysis for Various Types of Experiment

    Reading/Sample kth

    Test kth

    ST/SS

    ST/SS

    ST/MS

    ST/MS


    Finally summary measurement statement and interval estimation uncertainty of a measured quantity

    Single Sample: Report

    Multiple Samples: Report

    Single Sample: Report

    Multiple Samples: Report

    Students t

    pdf

    Area = a/ 2

    Finally: Summary Measurement Statement and Interval EstimationUncertainty of A Measured Quantity

    Measurement Statement


    2145 391 aerospace engineering laboratory i

    Some Uses of Uncertainty

    • Interpretation of Experimental Result

    • Comparing Theory and Experiment

    • Comparing Two Models

    • Industry


    Some uses of uncertainty interpretation of experimental result

    y

    y

    y

    x

    x

    x

    Some Uses of Uncertainty: Interpretation of Experimental Result

    With uncertainty, at least we have some indications of how good – or precise – is the current experimental result.

    Without uncertainty, we cannot evaluate how good – or precise - is the current experimental result.


    Some uses of uncertainty comparing theory and experiment

    Model A

    Model A

    y

    y

    y

    Model A

    x

    x

    x

    Some Uses of Uncertainty: Comparing Theory and Experiment

    With uncertainty, in this case we see that they are consistent (within the limit of the uncertainty of the current experimental result).

    Without uncertainty, we cannot compare whether or not the theoretical result is consistent with the experiment result.

    • Note:

    • Often a theoretical result requires constants that must – or should - be determined from experiment.

    • As a result, there are uncertainty associated with a theoretical result also.

    • Therefore, there should be error bars (though not shown here) associated with theoretical result also.

    With uncertainty, in this case we see that they are not consistent (within the limit of the uncertainty of the current experimental result).


    Some uses of uncertainty comparing two models

    y

    y

    Model A

    Model A

    Model B

    Model B

    x

    x

    Some Uses of Uncertainty: Comparing Two Models

    Without uncertainty of the experimental result,

    we cannot differentiate which one, A or B, is better.

    With uncertainty of the experimental result,

    in this case we see that the performance of models A and B cannot be differentiated with the current experimental result.


    Some uses of uncertainty industry

    Some Uses of Uncertainty: Industry

    • Would you like to know – roughly:

    • How accurate (or uncertain) are the flowmeters at gas stations in Bangkok?

    • Let’s say, you pay ~ 1,000 bahts/week  52,000 bahts / year.

    • Is it + 10 % @ 95% CL,

    • + 5 % @ 95% CL,

    • + 1 % @ 95% CL,

    • + 0.5 % @ 95% CL?

    • Imagine, e.g., PTT who sells petrol/gas in billions of bahts.

    • Since we cannot avoid this uncertainty – but we can try to minimize it, what would you do if you were, e.g., PTT, and you were uncertain ~ + 10%, + 5%, + 1%, + 0.5% ?


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